---
title: "A solid insulating sphere of radius \\(R\\) contains a total net charge \\(Q\\). The volume charge density \\(\\rho\\) inside the sphere varies directly with radial distance \\(r\\) from the center, such that \\(\\rho(r) = C r\\) for \\(r \\le R\\), where \\(C\\) is a constant. In terms of \\(Q\\), \\(R\\), \\(r\\), and fundamental constants, which of the following expressions represents the magnitude of the electric field at a distance \\(r < R\\) from the center of the sphere?"
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url: "https://nerd-notes.com/ubq/117948/"
date_modified: "2026-08-04T08:02:40+00:00"
---

# A solid insulating sphere of radius \(R\) contains a total net charge \(Q\). The volume charge density \(\rho\) inside the sphere varies directly with radial distance \(r\) from the center, such that \(\rho(r) = C r\) for \(r \le R\), where \(C\) is a constant. In terms of \(Q\), \(R\), \(r\), and fundamental constants, which of the following expressions represents the magnitude of the electric field at a distance \(r < R\) from the center of the sphere?

A solid insulating sphere of radius \(R\) contains a total net charge \(Q\). The volume charge density \(\rho\) inside the sphere varies directly with radial distance \(r\) from the center, such that \(\rho(r) = C r\) for \(r \le R\), where \(C\) is a constant. In terms of \(Q\), \(R\), \(r\), and fundamental constants, which of the following expressions represents the magnitude of the electric field at a distance \(r < R\) from the center of the sphere?

![A cross-sectional view of a solid sphere of radius R centered at the origin. The sphere is shaded with a radial gradient that becomes progressively darker toward the outer edge at radius R, representing an increasing volume charge density. A concentric dashed circular line inside the sphere of radius r (where r < R) represents a Gaussian surface. A radial arrow extends from the center to the dashed surface labeled r, and a second radial arrow extends from the center to the outer edge labeled R. A vector arrow labeled E points radially outward from a point on the dashed Gaussian surface. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830559-CQDe3R.jpg)

- **A.** \(\dfrac{Q r^2}{4\pi \varepsilon_0 R^4}\)
- **B.** \(\dfrac{Q r}{4\pi \varepsilon_0 R^3}\)
- **C.** \(\dfrac{3 Q r^2}{16 \pi \varepsilon_0 R^4}\)
- **D.** \(\dfrac{Q r^3}{4\pi \varepsilon_0 R^5}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117948/*